Hilbert’s Conception of “Existence-in-Mathematics” and Modelling it by a Formal Axiomatic Theory Ф+⁆, Treating Existence not as the Quanifier but as a Modality
DOI:
https://doi.org/10.47850/RL.2024.5.1.16-50Abstract
The subject-matter of investigation is Hilbert’s-principle consisting of two parts: (a) proclaiming equivalence-of-consistency-and-truth-in-mathematics; (b) proclaiming equivalence-of-consistency-and-existence-in‑mathematics. The target – explication and vindication of the principle. The scientific novelty: for reaching the goal,
(1) a hitherto unknown logically formalized multimodal axiomatic epistemology-and-ontology-system called Ф+⁆ has been constructed; (2) by means of Ф+⁆ , a precise axiomatic definition of the notion “existence as modality” is submitted for the first time; (3) by means of artificial language of Ф+⁆ , a precise formulation of the principle-of-equivalence-of-consistency-and-existence is given; (4) for the first time, the article presents formal deductive inferences (in formal-theory-Ф+⁆ ) of such formulae which make up Hilbert’s principle (given an appropriate interpretation of these formulae).
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